K11a53

K11a52

K11a54

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	K11a53 Quick Notes

K11a53 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X8394 X14,6,15,5 X16,7,17,8 X2,9,3,10 X18,11,19,12 X20,13,21,14 X22,16,1,15 X10,17,11,18 X12,19,13,20 X6,21,7,22
Gauss code	1, -5, 2, -1, 3, -11, 4, -2, 5, -9, 6, -10, 7, -3, 8, -4, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code	4 8 14 16 2 18 20 22 10 12 6
Conway Notation	[.4.210]

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{2,3\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a53/ThurstonBennequinNumber Hyperbolic Volume 14.416 A-Polynomial See Data:K11a53/A-polynomial

[edit Notes for K11a53's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant 4

[edit Notes for K11a53's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-6t^{3}+14t^{2}-18t+19-18t^{-1}+14t^{-2}-6t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+2z^{6}-2z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 97, -4 }
Jones polynomial	$-q+4-6q^{-1}+10q^{-2}-13q^{-3}+15q^{-4}-15q^{-5}+13q^{-6}-10q^{-7}+6q^{-8}-3q^{-9}+q^{-10}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{8}+3z^{2}a^{8}+a^{8}-2z^{6}a^{6}-8z^{4}a^{6}-8z^{2}a^{6}-2a^{6}+z^{8}a^{4}+5z^{6}a^{4}+8z^{4}a^{4}+5z^{2}a^{4}-z^{6}a^{2}-3z^{4}a^{2}+2a^{2}$
Kauffman polynomial (db, data sources)	$z^{4}a^{12}-z^{2}a^{12}+3z^{5}a^{11}-3z^{3}a^{11}+za^{11}+5z^{6}a^{10}-5z^{4}a^{10}+2z^{2}a^{10}+6z^{7}a^{9}-6z^{5}a^{9}+z^{3}a^{9}+za^{9}+6z^{8}a^{8}-8z^{6}a^{8}+4z^{4}a^{8}-3z^{2}a^{8}+a^{8}+5z^{9}a^{7}-9z^{7}a^{7}+7z^{5}a^{7}-6z^{3}a^{7}+za^{7}+2z^{10}a^{6}+4z^{8}a^{6}-26z^{6}a^{6}+32z^{4}a^{6}-16z^{2}a^{6}+2a^{6}+10z^{9}a^{5}-35z^{7}a^{5}+38z^{5}a^{5}-17z^{3}a^{5}+3za^{5}+2z^{10}a^{4}+2z^{8}a^{4}-29z^{6}a^{4}+38z^{4}a^{4}-12z^{2}a^{4}+5z^{9}a^{3}-19z^{7}a^{3}+19z^{5}a^{3}-6z^{3}a^{3}+2za^{3}+4z^{8}a^{2}-16z^{6}a^{2}+16z^{4}a^{2}-2z^{2}a^{2}-2a^{2}+z^{7}a-3z^{5}a+z^{3}a$
The A2 invariant	$q^{30}-q^{26}+q^{24}-2q^{22}+2q^{20}-q^{18}-q^{16}+q^{14}-4q^{12}+3q^{10}-q^{8}+2q^{6}+2q^{4}+2-q^{-2}$
The G2 invariant	$q^{162}-2q^{160}+4q^{158}-6q^{156}+5q^{154}-4q^{152}-2q^{150}+11q^{148}-20q^{146}+28q^{144}-30q^{142}+22q^{140}-4q^{138}-21q^{136}+50q^{134}-68q^{132}+71q^{130}-56q^{128}+20q^{126}+22q^{124}-60q^{122}+91q^{120}-96q^{118}+87q^{116}-61q^{114}+18q^{112}+33q^{110}-87q^{108}+125q^{106}-133q^{104}+99q^{102}-29q^{100}-57q^{98}+125q^{96}-140q^{94}+94q^{92}-100q^{88}+143q^{86}-110q^{84}+9q^{82}+119q^{80}-205q^{78}+213q^{76}-129q^{74}-20q^{72}+170q^{70}-272q^{68}+278q^{66}-195q^{64}+47q^{62}+107q^{60}-213q^{58}+249q^{56}-202q^{54}+87q^{52}+43q^{50}-151q^{48}+181q^{46}-129q^{44}+16q^{42}+113q^{40}-183q^{38}+167q^{36}-68q^{34}-73q^{32}+199q^{30}-249q^{28}+203q^{26}-79q^{24}-66q^{22}+181q^{20}-216q^{18}+181q^{16}-90q^{14}-4q^{12}+72q^{10}-101q^{8}+91q^{6}-55q^{4}+21q^{2}+7-18q^{-2}+17q^{-4}-14q^{-6}+7q^{-8}-3q^{-10}+q^{-12}$

Further Quantum Invariants

K11a53 Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a53"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$t^{4}-6t^{3}+14t^{2}-18t+19-18t^{-1}+14t^{-2}-6t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+2z^{6}-2z^{4}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 97, -4 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$-q+4-6q^{-1}+10q^{-2}-13q^{-3}+15q^{-4}-15q^{-5}+13q^{-6}-10q^{-7}+6q^{-8}-3q^{-9}+q^{-10}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$z^{4}a^{8}+3z^{2}a^{8}+a^{8}-2z^{6}a^{6}-8z^{4}a^{6}-8z^{2}a^{6}-2a^{6}+z^{8}a^{4}+5z^{6}a^{4}+8z^{4}a^{4}+5z^{2}a^{4}-z^{6}a^{2}-3z^{4}a^{2}+2a^{2}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$z^{4}a^{12}-z^{2}a^{12}+3z^{5}a^{11}-3z^{3}a^{11}+za^{11}+5z^{6}a^{10}-5z^{4}a^{10}+2z^{2}a^{10}+6z^{7}a^{9}-6z^{5}a^{9}+z^{3}a^{9}+za^{9}+6z^{8}a^{8}-8z^{6}a^{8}+4z^{4}a^{8}-3z^{2}a^{8}+a^{8}+5z^{9}a^{7}-9z^{7}a^{7}+7z^{5}a^{7}-6z^{3}a^{7}+za^{7}+2z^{10}a^{6}+4z^{8}a^{6}-26z^{6}a^{6}+32z^{4}a^{6}-16z^{2}a^{6}+2a^{6}+10z^{9}a^{5}-35z^{7}a^{5}+38z^{5}a^{5}-17z^{3}a^{5}+3za^{5}+2z^{10}a^{4}+2z^{8}a^{4}-29z^{6}a^{4}+38z^{4}a^{4}-12z^{2}a^{4}+5z^{9}a^{3}-19z^{7}a^{3}+19z^{5}a^{3}-6z^{3}a^{3}+2za^{3}+4z^{8}a^{2}-16z^{6}a^{2}+16z^{4}a^{2}-2z^{2}a^{2}-2a^{2}+z^{7}a-3z^{5}a+z^{3}a$

Vassiliev invariants

V2 and V3:

(0, 2)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $0$ $16$ $0$ $-48$ $16$ $0$ $-{\frac {32}{3}}$ $-{\frac {224}{3}}$ $-80$ $0$ $128$ $0$ $0$ $680$ $-{\frac {608}{3}}$ ${\frac {2816}{3}}$ ${\frac {152}{3}}$ $104$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$-4 is the signature of K11a53. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 3                       1 -1 1                     3   3 -1                   3 1   -2 -3                 7 3     4 -5               7 4       -3 -7             8 6         2 -9           7 7           0 -11         6 8             -2 -13       4 7               3 -15     2 6                 -4 -17   1 4                   3 -19   2                     -2 -21 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-5$ $i=-3$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[11, Alternating, 53]]
Out[2]=	11
In[3]:=	PD[Knot[11, Alternating, 53]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[14, 6, 15, 5], X[16, 7, 17, 8], X[2, 9, 3, 10], X[18, 11, 19, 12], X[20, 13, 21, 14], X[22, 16, 1, 15], X[10, 17, 11, 18], X[12, 19, 13, 20], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 53]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -11, 4, -2, 5, -9, 6, -10, 7, -3, 8, -4, 9, -6, 10, -7, 11, -8]
In[5]:=	BR[Knot[11, Alternating, 53]]
Out[5]=	BR[Knot[11, Alternating, 53]]
In[6]:=	alex = Alexander[Knot[11, Alternating, 53]][t]
Out[6]=	-4 6 14 18 2 3 4 19 + t - -- + -- - -- - 18 t + 14 t - 6 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 53]][z]
Out[7]=	4 6 8 1 - 2 z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 53]}
In[9]:=	{KnotDet[Knot[11, Alternating, 53]], KnotSignature[Knot[11, Alternating, 53]]}
Out[9]=	{97, -4}
In[10]:=	J=Jones[Knot[11, Alternating, 53]][q]
Out[10]=	-10 3 6 10 13 15 15 13 10 6 4 + q - -- + -- - -- + -- - -- + -- - -- + -- - - - q 9 8 7 6 5 4 3 2 q q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 53]}
In[12]:=	A2Invariant[Knot[11, Alternating, 53]][q]
Out[12]=	-30 -26 -24 2 2 -18 -16 -14 4 3 2 + q - q + q - --- + --- - q - q + q - --- + --- - 22 20 12 10 q q q q -8 2 2 2 q + -- + -- - q 6 4 q q
In[13]:=	Kauffman[Knot[11, Alternating, 53]][a, z]
Out[13]=	2 6 8 3 5 7 9 11 2 2 -2 a + 2 a + a + 2 a z + 3 a z + a z + a z + a z - 2 a z - 4 2 6 2 8 2 10 2 12 2 3 3 3 12 a z - 16 a z - 3 a z + 2 a z - a z + a z - 6 a z - 5 3 7 3 9 3 11 3 2 4 4 4 17 a z - 6 a z + a z - 3 a z + 16 a z + 38 a z + 6 4 8 4 10 4 12 4 5 3 5 32 a z + 4 a z - 5 a z + a z - 3 a z + 19 a z + 5 5 7 5 9 5 11 5 2 6 4 6 38 a z + 7 a z - 6 a z + 3 a z - 16 a z - 29 a z - 6 6 8 6 10 6 7 3 7 5 7 26 a z - 8 a z + 5 a z + a z - 19 a z - 35 a z - 7 7 9 7 2 8 4 8 6 8 8 8 3 9 9 a z + 6 a z + 4 a z + 2 a z + 4 a z + 6 a z + 5 a z + 5 9 7 9 4 10 6 10 10 a z + 5 a z + 2 a z + 2 a z
In[14]:=	{Vassiliev[2][Knot[11, Alternating, 53]], Vassiliev[3][Knot[11, Alternating, 53]]}
Out[14]=	{0, 2}
In[15]:=	Kh[Knot[11, Alternating, 53]][q, t]
Out[15]=	4 7 1 2 1 4 2 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 7 17 7 17 6 15 6 15 5 q q q t q t q t q t q t q t 4 7 6 8 7 7 8 6 ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 13 5 13 4 11 4 11 3 9 3 9 2 7 2 7 q t q t q t q t q t q t q t q t 2 7 3 t 3 t t 2 3 3 ---- + --- + --- + -- + 3 q t + q t 5 3 q q q t q